Langlands classification: Difference between revisions

In [[mathematics]], the '''Langlands classification''' is a classificationdescription of the [[irreducible representation]]s of a reductive [[Lie group]] ''G'', suggested by [[Robert Langlands]] (1973). MoreThere precisely,are ittwo classifiesslightly different versions of the Langlands classification. One of these describes the irreducible [[Admissible representation|admissible]] (''g'', ''K'')-[[module (mathematics)|module]]s,

for ''g'' a [[Lie algebra]] of a reductive Lie group ''G'', with [[maximal compact subgroup]] ''K'', in terms of [[tempered representation]]s of smaller groups. The tempered representations were in turn classified by [[Anthony W. Knapp|Anthony Knapp]] and [[Gregg Zuckerman]]. The other version of the Langlands classification divides the irreducible representations into [[L-packet]]s, and classifies the L-packets in terms of certain homomorphisms of the [[Weil group]] of '''R''' or '''C''' into the [[Langlands dual group]].

The Langlands classification states that the irreducible [[admissible representation]]s of (''g'', ''K'') are parameterized by triples

:(''F'', σσ,λ λ)

*''Q'' is the standard [[Borel subgroup|parabolic subgroup]] of ''F'', with [[Langlands decomposition]] ''Q'' = ''MAN''

*λ is an element of Hom(''a''_''F'', '''C''') with α(Re(λ)) > 0 for all simple roots α not in ''F''.

*{{Citation | last1=Adams | first1=Jeffrey | last2=Barbasch | first2=Dan | last3=Vogan | first3=David A. | title=The Langlands classification and irreducible characters for real reductive groups | url=httphttps://books.google.com/books?id=T37ryFaTWm4C | publisher=Birkhäuser Boston | location=Boston, MA | series=Progress in Mathematics | isbn=978-0-8176-3634-0 | mr=1162533 | year=1992 | volume=104}}

*E. P. van den Ban, ''Induced representations and the Langlands classification,'' in ISBN {{isbn|0-8218-0609-2}} (T. Bailey and A. W. Knapp, eds.).

* [[Armand Borel|Borel, A.]] and [[Nolan Wallach|Wallach, N.]] ''Continuous cohomology, discrete subgroups, and representations of reductive groups''. Second edition. Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000. xviii+260 pp. ISBN {{isbn|0-8218-0851-6}}

*{{Citation | last1=Langlands | first1=Robert P. | editor1-last=Sally | editor1-first=Paul J. | editor2-last=Vogan | editor2-first=David A. | title=Representation theory and harmonic analysis on semisimple Lie groups | origyearorig-year=1973 | chapter-url=http://publications.ias.edu/rpl/paper/16 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Math. Surveys Monogr. | isbn=978-0-8218-1526-7 | mr=1011897 | year=1989 | volume=31 | chapter=On the classification of irreducible representations of real algebraic groups | pages=101–170}}

*{{Citation | last1=Vogan | first1=David A. | editor1-last=Kobayashi | editor1-first=Toshiyuki | editor2-last=Kashiwara | editor2-first=Masaki | editor2-link=Masaki Kashiwara | editor3-last=Matsuki | editor3-first=Toshihiko | editor4-last=Nishiyama | editor4-first=Kyo | editor5-last=Oshima | editor5-first=Toshio | title=Analysis on homogeneous spaces and representation theory of Lie groups, Okayama--Kyoto (1997) | chapter-url=http://atlas.math.umd.edu/papers/kyoto.pdf | publisher=Math. Soc. Japan | location=Tokyo | series=Adv. Stud. Pure Math. | isbn=978-4-314-10138-7 | mr=1770725 | year=2000 | volume=26 | chapter=A Langlands classification for unitary representations | pages=299–324}}

*D. Vogan, ''Representations of real reductive Lie groups'', ISBN {{isbn|3-7643-3037-6}}